2.5.1. Local situation [00N0]

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2.5.1. Local situation

For affine varieties, the topological space of its analytification is defined in the same way as the spectrum of Banach algebra, except that boundedness requirement of seminorms are dropped. They enjoy similar basic properties as the spectrum of Banach algebra. Proofs are of same spirit hence are omitted.

Definition 2.83.

Let $Z=\spec(A_{Z})$ be an affine $k$ -scheme of finite type, where $A_{Z}$ is a $k$ -algebra of finite type. Its Berkovich analytification $Z^{an}$ is the topological space constituting of all multiplicative seminorms $\vvvert\mathord{\cdot}\vvvert$ on $A_{Z}$ as points and with the canonical topology (the weakest topology making every function $\vvvert\mathord{\cdot}\vvvert\to\vvvert f\vvvert$ continuous for each $f\in A_{Z}$ ). ([Ber, Remark 3.4.2])

Definition 2.84.

A character on $A_{Z}$ is a homomorphism of $k$ -algebra from $A_{Z}$ to some valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ over $(k,\lvert\mathord{\cdot}\rvert_{k})$ . Two characters $\chi_{1}:A_{Z}\to K_{1}$ and $\chi_{1}:A_{Z}\to K_{1}$ are called equivalent if there exists a $k$ -algebra homomorphism $\chi_{3}:A_{Z}\to K_{3}$ and norm preserving $k$ -algebra homomorphisms $i_{1}:K_{1}\to K_{3}$ and $i_{2}:K_{2}\to K_{3}$ satisfying $\chi_{3}=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$ .

Lemma 2.85.

There is a bijective map from the set of points of $Z^{an}$ to the set of equivalent classes of characters on $A_{Z}$ .

Proposition 2.86.

Let $\phi:A_{Z}\to A_{W}$ be a homomorphism of $k$ -algebras of finite type where $Z=\spec(A_{Z})$ and $W=\spec(A_{W})$ . Then there is an induced continuous map $\phi^{\star}:W^{an}\to Z^{an}$ , which sends a multiplicative seminorm $|\cdot|_{w}$ to $|\phi(\cdot)|_{w}$ .

Proposition 2.87.

If $\phi$ is surjective, then $\phi^{\star}$ is injective and is a closed map; if $\phi$ is finite, then $\phi^{\star}$ is surjective. ( [Ber, Proposition 3.46 (6)(7)])

Proposition 2.88.

Let $Z=\spec(A_{Z})$ be an affine $k$ -variety, $\vvvert\mathord{\cdot}\vvvert$ an algebra norm on $A_{Z}$ and $\mathcal{A}_{Z}$ be the $k$ -Banach algebra obtained by completing $A_{Z}$ with respect to $\vvvert\mathord{\cdot}\vvvert$ . Then the canonical homomorphism of $k$ -algebras from $A_{Z}$ to $\mathcal{A}_{Z}$ induces a continuous map which embeds the Berkovich spectrum $\mathfrak{M}(\mathcal{A}_{Z})$ into $Z^{an}$ as a compact subspace (and is closed since $Z^{\mathrm{an}}$ is Hausdorff), and the Berkovich topology coincides with the induced topology from $Z^{an}$ .

Proof.

For any $z\in\mathfrak{M}(\mathcal{A}_{Z})$ , the multiplicative algebra seminorm (or the corresponding character) $\chi_{z}$ on $\mathcal{A}_{Z}$ corresponds to a unique multiplicative algebra seminorm on $A_{Z}$ by restriction. Since $A_{Z}$ is dense in $\mathcal{A}_{Z}$ , the family of open sets $\{U(f;p,q),\text{ }f\in A_{Z},\text{ }p,q\in\mathbb{R}\}$ form a basis for topology on $\mathfrak{M}(\mathcal{A}_{Z})$ , hence the inherited topology coincides with the originial topology. So the embedding is continuous, and the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is compact in $Z^{an}$ . Since the topology on $Z^{an}$ is Hausdorff, the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is closed. ∎

Definition 2.89.

An analytic function on open set $U\subseteq(\spec A_{Z})^{an}$ is a map $h:U\to\coprod_{z\in U}\hat{\kappa}(z)$ which is a local uniform limit of rational functions: every $z\in U$ has an open neighbourhood $U^{\prime}\subseteq U$ such that for every $\epsilon>0$ , there exists $f_{U^{\prime}},g_{U^{\prime}}\in A_{Z}$ with $|h(z)-\frac{f_{U^{\prime}}(z)}{g_{U^{\prime}}(z)}|<\epsilon$ and $g(z)\neq 0$ for all $z\in U^{\prime}$ . Denote by $\mathcal{R}^{\mathrm{an}}(U)$ the $k$ -algebra of all analytic functions on $U$ .

Definition 2.90.

The structural sheaf $\mathscr{O}_{Z^{\mathrm{an}}}$ on $Z^{\mathrm{an}}$ is the one assigning $\mathcal{R}^{\mathrm{an}}(U)$ to an open set $U$ .

Proposition 2.91.

$\mathscr{O}_{Z^{\mathrm{an}}}$ is a sheaf of local rings. The pair $(Z^{\mathrm{an}},\mathscr{O}_{Z^{\mathrm{an}}})$ gives rise to a locally ringed space.

Proposition 2.92.

If $\mathcal{A}_{Z}$ is an affinoid algebra $\mathcal{A}_{Z}$ , then there is a morphism of locally ringed space

(\mathfrak{M}(\mathcal{A}_{Z},\mathscr{O}_{\mathfrak{M}(\mathcal{A}_{Z})})\to(Z^{an},\mathscr{O}_{Z^{an}})

Proof.

The map of topological spaces is given in Proposition 2.88. For the ring homomorphism, it suffices to construct a $k$ -algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$ for any open set $U\subseteq\mathfrak{M}(\mathcal{A})$ and any affinoid domain $V\subseteq U$ . Moreover, it suffices to consider $U$ and $V$ of basic form

U=U(\underline{p}^{-1}\underline{f},\underline{q}\underline{g}^{-1}),\quad V=\mathfrak{M}(\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1}))\text{ , }\epsilon>0

There is a homomorphism of $k$ -algebras $\mathcal{R}^{\mathrm{an}}(U)\to\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1})$ sending $\frac{\tilde{f}}{\tilde{g}}$ for $\tilde{f},\tilde{g}\in A_{Z}$ to itself, the later being an element of $\mathcal{A}_{V}$ since $\frac{1}{\tilde{g}}\in\mathcal{A}_{V}$ by Lemma 2.25. As uniform limits of sequence in $\mathcal{R}^{\mathrm{an}}(U)$ remains to be uniform limits, this homomorphism extends to a $k$ -algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$ . ∎